Let \(p\)  be fixed. Let \(\alpha \in \mathbb {R}\)  be such that \( \mathcal {C}_{\alpha } := \lbrace  \mathbf {q}_A \in \mathcal {P}_1(A) , \mathcal {L}(\mathbf {q}_A,p) \le \alpha \rbrace  \)  is not empty. From the growth assumption on \(b\)  and \(f\) , we deduce the existence of a positive constant \(M_{\alpha }\)  such that, for all \( \mathbf {q}_A \in \mathcal {C}_{\alpha } \)  : \(\int _A |a|^{1+\delta } \mathbf {q}_A (da) \le M_{\alpha }\) . Now let \((\mathbf {q}_A^n) \in \mathcal {C}_{\alpha } \)  be a sequence converging in \(\mathcal {P}_1(A)\)  to \(\mathbf {q}_A\) . From [1]}, Lemma 5.1.7, using the fact that \(f\)  is lower semi-continuous and bounded from below and \(b\)  is continuous and grows at most linearly :
\(\left\lbrace \begin{array}{ll}\displaystyle \liminf _n \int _A f(a) \mathbf {q}_A^n (da) \ge \int _A f(a) \mathbf {q}_A (da) \\\displaystyle \lim _n \int _A b(a) \mathbf {q}_A^n (da) = \int _A b(a) \mathbf {q}_A (da).\end{array}\right.\) 
